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American Journal of Physics

Numerical implementation of Einstein-Brillouin-Keller quantization for arbitrary potentials

Andrew J. Larkoski, David G. Ellis, Lorenzo J. Curtis

Published Jul 1, 2006• 18 citations
Publisher

Abstract

The Einstein-Brillouin-Keller (EBK) quantization equation is used to determine the energy levels of a two-body system with an arbitrary central potential that allows for bound states. The treatment is based on the conservation laws and avoids both the Newtonian and Schrödinger differential equations. Because analytic solutions for the energy levels do not exist in general, the EBK condition is applied using the Newton-Raphson method and the radial probability density is computed. Potentials appropriate for a diatomic molecule are considered and the effect of the angular momentum on the radial distribution, the nature of the classical orbits, and the possibility of closed orbits is studied.

DOI
10.1119/1.2192788

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The Einstein-Brillouin-Keller (EBK) quantization equation is used to determine the energy levels of a two-body system with an arbitrary central potential that allows for bound states. The treatment is based on the conservation laws and avoids both the Newtonian and Schrödinger differential equations. Because analytic solutions for the energy levels do not exist in general, the EBK condition is applied using the Newton-Raphson method and the radial probability density is computed. Potentials appropriate for a diatomic molecule are considered and the effect of the angular momentum on the radial distribution, the nature of the classical orbits, and the possibility of closed orbits is studied.
Abstract We explore the semi-classical Sommerfeld method of quantization as applied to states with nonzero angular momentum, and show that it leads to qualitatively and quantitatively useful information about systems with spherically symmetric potentials. We begin by reviewing the traditional application of this model to hydrogen, and discuss the way Einstein–Brillouin–Keller (EBK) quantization resolves a mismatch between Sommerfeld states and true quantum mechanical states. We then analyze systems with logarithmic and Yukawa potentials, and compare the results of Sommerfeld and EBK quantization to those from solving Schrödinger’s equation. We show that the semi-classical quantization techniques provide insight into the spread of energy levels associated with a given principle quantum number, as well as giving quantitatively accurate approximations for the energies. We also argue that analyzing systems in this manner involves an interesting application of numerical methods, as well as providing insight into the connections between classical and quantum mechanical physics.
We examine the semiclassical limit of the quantum energy spectrum in many dimensions: by means of a WKB-like ansatz leading to Einstein-Brillouin-Keller (EBK) quantization, by means of a path integral, hence associating a bound state with a particular classical periodic trajectory, and by the Birkhoff-Gustavson (BG) transformation to action-angle variables. We extend the EBK method to many-fermion systems using coherent states; and apply both EBK using surfaces of section, and the BG transformation to an SU(3) schematic nuclear shell model. We describe a new algorithm for finding periodic trajectories of a Lagrangian system with polynomial potential. It is applied to the Henon-Heiles system with good results, and these trajectories are used to quantize the system. The EBK and BG methods have some success, while periodic trajectory quantization fails. We discuss possible reasons for this failure and future approaches to these problems.
We explore the method of old quantization as applied to states with nonzero angular momentum, and show that it leads to qualitatively and quantitatively useful information about systems with spherically symmetric potentials. We begin by reviewing the traditional application of this model to hydrogen, and discuss the way Einstein-Brillouin-Keller quantization resolves a mismatch between old quantization states and true quantum mechanical states. We then analyze systems with logarithmic and Yukawa potentials, and compare the results of old quantization to those from solving Schrodinger's equation. We show that the old quantization techniques provide insight into the spread of energy levels associated with a given principal quantum number, as well as giving quantitatively accurate approximations for the energies. Analyzing systems in this manner involves an educationally valuable synthesis of multiple numerical methods, as well as providing deeper insight into the connections between classical and quantum mechanical physics.
Abstract The energy levels of systems whose classical motion is multiply periodic are accurately given by the quantum conditions of Einstein, Brillouin & Keller (E. B. K.). We transform the E. B. K. conditions into a representation of the spectrum in terms of a ‘topological sum’ involving only the closed classical orbits; the theory applies equally to separable and non-separable systems; stability parameters are not involved. Significant contributions come from complex closed orbits which however have real constants of the motion. Clustering of levels on different scales is demonstrated by smoothing the spectrum using the formal device, due to Balian & Bloch, of adding a variable imaginary part to the energy. The topological sum is shown to agree very well with exactly-computed spectra for circular and spherical potential wells with repulsive core.
A method for Einstein-Brillouin-Keller (EBK) semiclassical wave functions is applied to the case where the quantizing torus is close to a chain of resonance islands, and standard EBK wave functions are distorted by the resonance islands. The method tested here is a modification of a recent ``direct trajectory'' implementation of the standard EBK method. The phase factor of the semiclassical wave function is obtained exactly as by Yang and Kellman [Phys. Rev. A 62, 022105 (2000); 65, 034103 (2002)], by running a single trajectory on the quantizing torus. However, rather than obtaining the amplitude factor from this trajectory, the method here uses an ensemble of trajectories uniformly distributed in a region enclosing an area \ensuremath{\Elzxh} around the torus. The agreement between the resulting semiclassical wave function and the quantum wave function is excellent.
The Schrödinger-Newton (S-N) equations were proposed by Penrose [18] as a model for gravitational collapse of the wave-function. The potential in the Schrödinger equation is the gravity due to the density of ∣ψ∣2|\psi|^2∣ψ∣2, where ψ\psiψ is the wave-function. As with normal Quantum Mechanics the probability, momentum and angular momentum are conserved. We first consider the spherically symmetric case, here the stationary solutions have been found numerically by Moroz et al [15] and Jones et al [3]. The ground state which has the lowest energy has no zeros. The higher states are such that the (n+1)(n+1)(n+1)th state has nnn zeros. We consider the linear stability problem for the stationary states, which we numerically solve using spectral methods. The ground state is linearly stable since it has only imaginary eigenvalues. The higher states are linearly unstable having imaginary eigenvalues except for nnn quadruples of complex eigenvalues for the (n+1)(n+1)(n+1)th state, where a quadruple consists of {λ,λˉ,−λ,−λˉ}\{\lambda,\bar{\lambda},-\lambda,-\bar{\lambda}\}{λ,λˉ,−λ,−λˉ}. Next we consider the nonlinear evolution, using a method involving an iteration to calculate the potential at the next time step and Crank-Nicolson to evolve the Schrödinger equation. To absorb scatter we use a sponge factor which reduces the reflection back from the outer boundary condition and we show that the numerical evolution converges for different mesh sizes and time steps. Evolution of the ground state shows it is stable and added perturbations oscillate at frequencies determined by the linear perturbation theory. The higher states are shown to be unstable, emitting scatter and leaving a rescaled ground state. The rate at which they decay is controlled by the complex eigenvalues of the linear perturbation. Next we consider adding another dimension in two different ways: by considering the axisymmetric case and the 2-D equations. The stationary solutions are found. We modify the evolution method and find that the higher states are unstable. In 2-D case we consider rigidly rotationing solutions and show they exist and are unstable.